A mahāvidyā inference is used for establishing another inference. Its Reason is normally an omnipresent property. Its Target is defined in terms of a general feature that is satisfied by different properties in different cases. It assumes that there is no case that has the absence of its Target. The main defect of a mahāvidyā inference μ is a counterbalancing inference that can be formed by a little modification of μ. The discovery of its counterbalancing inference can invalidate such an (...) inference. This paper will argue that Cantor’s diagonal argument too shares some features of the mahāvidyā inference. A diagonal argument has a counterbalanced statement. Its main defect is its counterbalancing inference. Apart from presenting an epistemological perspective that explains the disquiet over Cantor’s proof, this paper would show that both the mahāvidyā and diagonal argument formally contain their own invalidators. (shrink)

Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.

Conceptual limitations restrict our epistemic options. One cannot believe, disbelieve, or doubt what one cannot grasp. I show how, even granting an epistemic ought-implies-can principle, such restrictions might lead to epistemic dilemmas: situations where each of one’s options violates some epistemic requirement. An alternative reaction would be to take epistemic norms to be sensitive to one’s options in ways that ensure dilemmas never arise. I propose, on behalf of the dilemmist, that we treat puzzlement as a kind of epistemic residue, (...) roughly analogous with guilt, appropriate only when one has violated an epistemic requirement. Sometimes, in bumping up against the limits of one’s concepts, it is appropriate to be puzzled no matter what one believes. Puzzlement can thus play the same role in an epistemic dilemmist’s theory that guilt plays in the theories of many moral dilemmists. (shrink)

Samuel J. Wheeler defends Wittgenstein's criticism of Cantor's set theory against the objections raised by Hilary Putnam. Putnam claims that Wittgenstein's dismissal of the basic tenets of this set theory concerning the noncountability of the set of real numbers was unfounded and ill‐conceived. In Wheeler's view, Putnam's charges result from his failure to grasp Wittgenstein's intention and, in particular, to consider the difference between empirical and logical impossibility. In my paper, I argue that Wheeler's defence is unsuccessful and, at the (...) same time, that Putnam's objections go too far. (shrink)

This paper expands upon a way in which we might rationally doubt that there are multiple sizes of infinity. The argument draws its inspiration from recent work in the philosophy of truth and philosophy of set theory. More specifically, elements of contextualist theories of truth and multiverse accounts of set theory are brought together in an effort to make sense of Cantor’s troubling theorem. The resultant theory provides an alternative philosophical perspective on the transfinite, but has limited impact on everyday (...) mathematical practice. (shrink)

The paper claims that the strategy adopted in the proof of Cantor’s theorem is problematic. Using the strategy, an unacceptable situation is built. The paper also makes the suggestion that the proof of Cantor’s theorem is possible due to lack of an apparatus to represent emptiness at a certain level in the ontology of set-theory.

Who gets to contribute to knowledge production of an epistemic community? Scholarship has focussed on unjustified forms of exclusion. Here I study justified forms of exclusion by investigating the phenomenon of so-called ‘cranks’ in mathematics. I argue that workload-management concerns justify the exclusion of these outsiders from mathematical knowledge-making practices. My discussion reveals three insights. There are reasons other than incorrect mathematical argument that justify exclusions from mathematical practices. There are instances in which mathematicians are justified in rejecting even correct (...) mathematical arguments. Finally, the way mathematicians spot mathematical crankery does not support the pejorative connotations of the ‘crank’ terminology. (shrink)